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By David William Koster

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Further reproduction prohibited without permission. 22 (The s u b s c r i p t on th e l e t t e r d e n o tin g t h e graph i s one l e s s th a n th e number o f v e r t i c e s . ) L ist 2 O -o , 6 6 0 3 o—-— 0 - 7 - 0 4 p 4 : O , O------ O , O-------O , O----—O , O-------O , O------ O 6 4 4 4 6 6 3 6 3 8 4 12 3 4 (p ^ 3) , o-------o—— •o , o------- o-----—o , o----- o—— o 6 4 4 4 4 4 4 O------- 0 -- 7— 0 3 3 3 -o , o--------o------o3 3 3 4 -a—r— o------ o * F. o4 E6 E8 B^’ ^ ° o o o- , o— — o------- o— -— o 3 o3 3 3 4 _o------- o------ o------ o 3 3 3 3 — o-------o ^ o o , -o o -o o o o o o o— o ( i ^ 2) o——o—-—o , O—;—o------o o , O-r-—o---- o 4 p i' ' p 4 4 4 P P A ■ o Reproduced with permission of the copyright owner.

0 Now o u r r e p r e s e n t a ­ Let Fe C+ be c o n n e c te d . I f W = W(P) i s f i n i t e , th e n i s a f a i t h f u l r e p r e s e n t a t i o n o f W. P roof: By t h e remarks p r e c e d in g t h e c o r o l l a r y we can assume t h a t n o t a l l p ^ , 1 < i < H, a r e 2. L i s t 1. in But a g lan ce a t L i s t 1 r e v e a l s t h e f a c t t h a t s in c e n o t a l l p^ a r e 2 , r must be l i n e a r . [3] Now s i n c e W i s f i n i t e , r a p p e a rs As rem ark ed b e f o r e , f o r su ch P , C o x eter g iv e s a r e p r e s e n t a t i o n o f W w hich i s th e r e p r e s e n t a t i o n p o f P rop­ o s it i o n 3.

1 = th e r e f le c tin g and e£ = e ( A j ) . Again we and th u s p . l e . s o i n p a r t i c u l a r , p. < e . r il x r ri — x F o r (l)we a p p ly Lemma 3 to g e t }®/g ' | “ 2. P r o p o s i t i o n 9 c e r t a i n l y im p lie s t h a t | g/ g ' | | G/ G '| > e L > p ]_ = We th u s o b t a i n t h e sequence 2= | G/g . | . Hence, e^ = p^ and A ^ Is t h e o n ly o r b i t . For (2) th e argum ent i s t h e same as f o r ( 1 ) . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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Complex reflection groups by David William Koster


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